Analytical vs Experimental Determination of Thermal Resistance with Scalable DC HEMT Model

Analytical vs Experimental Determination of Thermal Resistance with Scalable DC HEMT Model

Sagar Karalkar and Kanti Prasad Electrical & Computer Engineering University of Massachusetts Lowell Lowell, MA 01854, USA

Yu Zhu, Jerod Mason, and Dylan Bartle Skyworks Solutions Inc.

Woburn, MA 01801, USA

Abstract This paper presents a new approach for extracting thermal resistance analytically and comparing it with experi- mental data along with scalable DC HEMT models for devices with varying Nf and Ugw. Analytical approach to predict the devices temperature increase distribution and calculate the thermal resistance with the help of that is shown. Temperature profile across the fingers with varying devices is carried out and compared with the experimental data. In this way, the analytical values for thermal resistance obtained with the temperature distribution and thermal values used for fitting the output, transfer characteristics are very close to each other and a very good fitting of each device have been achieved.

Index terms HEMTs, thermal resistance, optimization, tun- ing, circuit simulation, equivalent circuit, semiconductor device modeling and switching application

1. INTRODUCTION

   The thermal property of solid state devices has been very important in designing of the circuit. GaAs IC suffer even more from thermal effects since the thermal conductivity of GaAs is very less to that of silicon. HEMT is widely used in monolithic microwave integrated circuit (MMIC). In order to achieve the best circuit performance, device size needs to be optimized. An accurate scaling model will allow circuit de- signers to get the desired performance by choosing the right size devices. With the increase in the power density of semi- conductor devices, a scalable model with scalable thermal resistance becomes critical for predicting the thermal perfor- mance of different sized devices in a broad, dynamic operat- ing range. [1]-[3].

   Many techniques have been reported for measuring the thermal resistance of the devices, which are IRM [4], and pulsed electrical method [5]-[7], liquid crystal thermography for determining the hot spot [8] and dc electrical method which is used for measuring the thermal resistance using temperature dependence of gate metal resistivity [9].

   In this paper, a novel approach is proposed for determina- tion of thermal resistance by analytical approach and compar- ing it with the experimental data. The temperature increase is analytically solved with the rectangular heat source and the compared the experimental temperature increase of each de- vice. The temperature behavior is seen across each finger of

   the device. The thermal resistance value used to fit the output and transfer characteristics of all the devices are compared with the analytical approach.

2. EXPERIMENT

   Pseudomorphic AlGaAs/InGaAs/GaAs HEMT with ground-signal-ground test pads were fabricated on semi insu- lating GaAs substrates. The temperature increase of each de- vice were measured with the help of QFI thermal imag- ing..The transfer and output performances were measured via probe station. Ten devices with different geometry were measured and the data was recorded for individual devices. The ten devices with different geometries are shown below in Table I.

   Nf	Ugw(µm)	Total Gate Width
   5	143	.715mm
   7	143	1mm
   14	143	2mm
   21	143	3mm
   24	143	3.4mm
   5	25	.125mm
   5	50	.25mm
   5	75	.375mm
   5	100	.5mm
   5	200	1mm

   TABLE I: Device Geometry

   In table I, Nf is the number of fingers and Ugw is the unit gate width. Five devices with Varying Nf and other five de- vices with varying Ugw are shown above. The output per- formance was measured in the bias range of Vg = -1 to 0V and Vd = 0 to 3 V, transfer performance was measured in the range of Vd = 1 to 3 V and Vg = -1 to 0.5V and Ig -Vg (diode output) was measured in the range of Vg -3 to 1.2V with Vd constant. The device self-heating measurement of each device was carried from 1 to 3V.

   Fig 1: Device 1mm (Varying Nf)

   Device (mm)	Voltage (V)	Power (W)	Temp measured Value (0C)
   0.715	1	0.1	89.8
   0.715	2	0.28	97.3
   0.715	3	0.36	127
   1	1	0.12	90.16
   1	2	0.34	112.3
   1	3	0.45	133.1
   2	1	0.15	91.5
   2	2	0.48	114.1
   2	3	0.69	152.4
   3	1	0.19	92.45
   3	2	0.72	120.8
   3	3	1.02	176.1
   3.4	1	0.23	95.89
   3.4	2	0.86	126.8
   3.4	3	1.23	184.3

   TABLE II: Measured Temperature (Devices with Varying Nf)

   Fig 2: Device 1mm (Varying Ugw)

   Device (mm)	Voltage (V)	Power (W)	Temp measured Value (0C)
   0.125	1	0.02	88.3
   0.125	2	0.052	95.7
   0.125	3	0.069	103.2
   0.25	1	0.03	89.45
   0.25	2	0.084	97.6
   0.25	3	0.108	107.8
   0.375	1	0.04	89.49
   0.375	2	0.116	103.22
   0.375	3	0.144	110.07
   0.5	1	0.055	90.184
   0.5	2	0.14	104.79
   0.5	3	0.18	114.63
   1	1	0.1	90.85
   1	2	0.4	110.77
   1	3	0.54	127.61

   TABLE III: Mesured Temperature (Devices with Varying Ugw)

3. ANALYTICAL APPROACH

   A rectangular heat source is located at some depth z, with corners of heat source are at (x1,y1,z),(x2,y1,z),(x1,y2,z) and (x2,y2,z). The elemental temperature increase (T) at point source (x,y,z) is given as [14]:

   T QT

   4. .k.r

   dx' dy'

   (a)

   T(r)

   QT b

   x2 ,

   y2 ,z

   bx1,

   y1,z

   s

   r (x x')2 ( y y')2 (z z )2

   4. .k

   – bx1,y2 ,z bx2 ,y1,z

   Where r is the distance between the heat source and the temperature distribution in the infinite medium at (x,y,z). The temperature distribution T(r) is obtained by integrating equa- tion (a) where QT is power per unit area

   x2 y2 Q

   Where QT is the power density per unit area, k is the thermal conductivity. Where QT is P / WL.

   To check the temperature increase distribution for each de- vice at the end of the substrate, we need to incorporate the boundary conditions [14] equation (e) for upper and (f) for

   T(r) T

   x y 4. .k.r

   (b)

   bottom surface of the substrate.

   1 1

   Q x2y2

   d (x)d (y)

   T (r)

   T(r) T

   4. .k x y

   x2 y2 z2

   (c)

   0

   z z0

   (e)

   1 1

   Where x = x – x, y = y – y, z = z z, x1 = x x1, y1 = y y1, x2 = x x2, y2 = y y2Integrating equation (c)

   T (r)(x, y, z d) 0

   (f)

   with respect to y

   With the help of method of images [14] shown in the figure3

   Q y2

   d (y)

   4. .k y

   T(r) T

   1

   x2 y2 z2

   T(r)

   QT

   4. .k

   logy

   x2 y2 z2 y2

   Y1

   (d)

   Now integrating equation d with respect to x i.e without solving for the limit for previous relation [14].

   Q x2

   T

   2 2 2

   T(r)

   4. .k

   log y

   x1

   x y

   z

   d(x)

   T(r)

   QT

   4. .k

   a

   b

   y2 x2

   y1 x1

   a(x, z) ztan-1 x x

   z

   b(x, y,z) xlog y

   • ylogx

   x2 y2 z2

   x2 y2 z2

   Fig3: Method of Images for determining the temp distribution

   -1 xy

   – ztan

   z x2 y2 z2

   The original heat source is located at some depth z, the mirror image is located the same z depth above the center line, which implies one is at + direction and the other one is at negative direction. In this study with the figure below, all

   From above equation we can see that term a does not

   have any y dependency so the whole term a can be neg- lected and we are left with only term b. Therefore the tem- perature distribution after integral value is given as

   the heat sources above the center line are negative distances, whereas the heat sources located below the center line are positive distances. The overall heat source addition at the center is names as 0 which is the addition of both the sources to obtain the boundary condition to be satisfied at the

   top surface. In order to satisfy the boundary condition at z = d where the temperature distribution tends to zero, 2 heat sources are added at distance 2d. The 2 heat sources (1) lo- cated at distance 2d are negative heat sources so that they cancel with the heat sources located at center, to get the tem- perature increase zero at distance d i.e. ( = 0 in figure). As a result of this the second boundary condition is satisfied, but due to this the first boundary condition is not satisfied. In order to satisfy the first boundary condition, we need to add two same negative heat sources at negative distance through mirror image. This procedure continues and is infinite long to satisfy both the conditions. Experiment was tried to carry out for varying n i.e. number of heat sources located at different distances. With n (no of heat sources) = 100 we can see it is as close to as zero at the bottom surface at z =d shown in the figures below. The temperature increase distribution shown is for horizontal axis i.e along the x axis in the infinite medium.

   Fig 4: varying n, z=0.01*10-6

   Fig 5: varying n, z=d=635*10^-6

   n	temp increase at bottom surface
   0	1.2492
   10	0.059611
   20	0.030532
   50	0.012394
   70	0.0088782
   80	0.0077753
   100	0.006228
   200	0.0031218

   Table IV: temperature increase with varying n

4. MULTI DEVICE OPTIMIZATION

   A number of DC I-V has been proposed for modeling of GaAs Mesfets [10]-[12]. Figure 6 shows the equivalent cir- cuit of HEMT used in this study. Modified Angelov model has been used to predict the dc behavior [13]

   Fig 6: Equivalent circuit model for HEMT

   The five-diode models each with varying nf and ugw are then extracted simultaneously by performing a multi-device optimization. Tentative scaling rules with unknown coeffi- cients are assumed for each size-dependent model parameter. The model parameters can then be represented with the scal- ing rules, and the same model parameters for different devic- es are thus correlated. Instead of the model parameters, the unknown coefficients inside the scaling rules are now defined as the optimal variables. The error function now is the sum of the difference between the measurement and simulation for all of the devices. Multi device optimizations were carried out for all the devices with varying Nf and varying Ugw i.e each of them had five sets of devices. Each of the devices had around 39 parameters, so there were five times parameters for each multi device simulations. The thermal resistance value was compared to value obtained from the experimental mea- surement of each device. The optimization technique used for multi-device optimization in this study is the random and gradient optimization. Study started with random optimiza- tion to have the initial values for fitting, and after that gra- dient optimization type has been used.

   Fig 7: Output, transfer and Ig-Vg characteristics for Gate Width 3.4mm (Multi Device Optimization varying Nf)

   Fig 8: Output, transfer and Ig-Vg characteristics for Gate Width 0.5mm (Multi Device Optimization varying Ugw)

   As shown in Figs 7 and 8, excellent matches between the measurement and simulation have been achieved for output, transfer and Ig-Vg characteristics. The blue line is the measured data and the red one is the simulated result.

   Gate Width (mm)			Rth (ohms)	Rs(ohms)	Rg (ohms)
   0.125	0.70	2.99	302.24	4.94	11.21
   0.25	0.70	2.99	195.01	2.78	11.32
   0.375	0.70	2.99	153.62	2.07	11.43
   0.5	0.70	2.99	126.01	1.71	11.55
   1	0.70	2.99	72.32	1.17	12

   TABLE V: Parameters extracted with multi device optimization with varying Ugw

   Gate Width (mm)			Rth (ohms)	Rs(ohms)	Rg (Ohms)
   0715	0.70	2.48	100.85	1.38	11.74
   1	0.70	2.48	82.51	1.17	12
   2	0.70	2.48	59.5	0.90	12.9
   3	0.70	2.48	51.83	0.81	13.8
   3.4	0.70	2.48	50.03	0.79	14.16

   TABLE VI: Parameters extracted with multi device optimization with vary- ing Nf

   Some extracted parameters from multi-device optimization are shown in Table V and VI. The size dependent parameter follows exactly the scaling rule, and the size independent parameter keep the same value for different devices.

5. RESULTS

The equation derived for temperature distribution is used to find the temperature distribution profile for all the devices at different voltages with varying Nf and Ugw. The heat source position as described earlier in the equation are used as the length and unit gate width parameters to define the value along with variable to define to Nf in Matlab. The values for x1, x2, y1, y2 and z are -0.2µm, 0.2µm, -71.5µm, 71.5µm and 0.01µm to keep all the parameters of the same unit, so that the length of the device is 0.4µm which is fixed and Ugw is 143µm which is dependable on the device Ugw size.

Fig 9: Temp distribution of device 1mm with Nf = 7 and

Ugw = 143 at 2V.

Fig 10: Temp distribution of device 1mm with Nf = 5 and Ugw = 200 at

2V.

Tables VII and VIII show the values of each device at different voltage. Power is calculated by the simple V*I, where current for each device is recorded while performing the experiment. The analytical value extracted for each de- vice at each voltage by varying Nf and Ugw is recorded. To the analytical value reference value of 850C is added in order to compare the values obtained with that of experiment, as the reference plate was at 850C while measuring the tempera- ture behavior of each device. Main concentration was in the linear region of the device till 2 V.

TABLE VII: Comparison between analytical and measured Temp value for devices with Varying Nf

Fig 11: Analytical vs Measured for 1mm device (varying Nf)

P= V * I

Rth = T / P

T = T room temp

TABLE IX: Rth Analytical Value (Varying Ugw)

TABLE X: Rth Analytical Value (Varying Nf)

TABLE VIII: Comparison between analytical and measured Temp value for devices with Varying Ugw

Fig 12: Analytical vs Measured for 1mm device (varying Ugw)

The thermal resistance is calculated with the power calculated for each device and is given as

Comparing the values of thermal resistance from the table V and VI obtained with the simulating data fo devices on ADS with the analytical values from table IX and X, one can see that the value obtained are close enough and the fit- ting for output, transfer characteristics of each device are close to each as shown in figures 7 and 8.

The scaling rule of thermal resistance proposed in [14]-[16] are used in the study.

Rth t (7)

kWL

Where t is substrate thickness, k is the thermal conductivity, W is gate width, and L is the length. The extracted gate width dependence of Rth is shown in Fig 13 and 14. For the devices with self-heating effect, the accurate dc performance predic- tion cannot be achieved without an accurate scalable thermal resistance.

Fig13: Gate width vs. Rth (multi-device optimization Varying Ugw)

Fig14: Gate width vs. Rth (multi-device optimization Varying Nf)

CONCLUSION

A novel approach to extract dc HEMT model has been pro- posed and demonstrated with accurate extraction of thermal resistance. The experimental data compared to the analytical data for accurate values of thermal resistance is observed to be very close. The temperature distribution (horizontally or vertically) along with the maximum temperature across the fingers in each devices observed analytically vs experimen- tally has been very close. Scaling rule for thermal resistance has been confirmed for both the sets of devices with varying Nf and Ugw.

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